Quantitative estimates in Beurling--Helson type theorems

TL;DR

This paper investigates how the norms of exponential functions with phase functions grow in certain Fourier spaces as the frequency increases, providing insights relevant to variable change problems in these spaces.

Contribution

It offers quantitative estimates on the growth of norms of exponential functions in $A_p$ spaces for $C^1$ functions, extending Beurling--Helson type theorems.

Findings

Established growth rate estimates for $ orm{e^{ila}}_{A_p}$ as $|la| oe$

Connected norm growth to smoothness properties of phase functions

Applied results to change of variable problems in Fourier spaces

Abstract

We consider the spaces $A_p(\mathbb T)$ of functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\fu{\f}=\{\fu{\f}(k), ~k \in \mathbb Z\}$ belongs to $l^p, ~1\leq p<2$. The norm on $A_p(\mathbb T)$ is defined by $\|f\|_{A_p}=\|\fu{\f}\nolinebreak\|_{l^p}$. We study the rate of growth of the norms $\|e^{i\lambda\varphi}\|_{A_p}$ as $|\lambda|\rightarrow \infty, ~\lambda\in\mathbb R,$ for $C^1$ -smooth real functions $\varphi$ on $\mathbb T$. The results have natural applications to the problem on changes of variable in the spaces $A_p(\mathbb T)$.